Average Error: 12.4 → 0.4
Time: 13.4s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)}}\right) - 4.5\]
\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5
↓
\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)}}\right) - 4.5
double code(double v, double w, double r) {
	return (((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5);
}
↓
double code(double v, double w, double r) {
	return (((3.0 + ((2.0 / r) / r)) - ((0.125 * (3.0 - (2.0 * v))) / ((1.0 - v) / ((fabs(w) * r) * (fabs(w) * r))))) - 4.5);
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.4

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
  2. Using strategy rm

  3. Applied *-un-lft-identity12.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{\color{blue}{1 \cdot \left(1 - v\right)}}\right) - 4.5\]

  4. Applied times-frac8.2

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1} \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}}\right) - 4.5\]

  5. Simplified8.2

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right)} \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}\right) - 4.5\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt36.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot \color{blue}{\left(\sqrt{r} \cdot \sqrt{r}\right)}}{1 - v}\right) - 4.5\]

  8. Applied add-sqr-sqrt36.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot \color{blue}{\left(\sqrt{r} \cdot \sqrt{r}\right)}\right) \cdot \left(\sqrt{r} \cdot \sqrt{r}\right)}{1 - v}\right) - 4.5\]

  9. Applied add-sqr-sqrt36.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\color{blue}{\left(\sqrt{w \cdot w} \cdot \sqrt{w \cdot w}\right)} \cdot \left(\sqrt{r} \cdot \sqrt{r}\right)\right) \cdot \left(\sqrt{r} \cdot \sqrt{r}\right)}{1 - v}\right) - 4.5\]

  10. Applied unswap-sqr36.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\color{blue}{\left(\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right) \cdot \left(\sqrt{w \cdot w} \cdot \sqrt{r}\right)\right)} \cdot \left(\sqrt{r} \cdot \sqrt{r}\right)}{1 - v}\right) - 4.5\]

  11. Applied unswap-sqr36.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\color{blue}{\left(\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right) \cdot \sqrt{r}\right) \cdot \left(\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right) \cdot \sqrt{r}\right)}}{1 - v}\right) - 4.5\]

  12. Simplified36.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\color{blue}{\left(\left|w\right| \cdot r\right)} \cdot \left(\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right) \cdot \sqrt{r}\right)}{1 - v}\right) - 4.5\]

  13. Simplified0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left|w\right| \cdot r\right) \cdot \color{blue}{\left(\left|w\right| \cdot r\right)}}{1 - v}\right) - 4.5\]
  14. Using strategy rm

  15. Applied associate-/r*0.4

    \[\leadsto \left(\left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)}{1 - v}\right) - 4.5\]
  16. Using strategy rm

  17. Applied clear-num0.4

    \[\leadsto \left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)}}}\right) - 4.5\]

  18. Applied un-div-inv0.4

    \[\leadsto \left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)}}}\right) - 4.5\]

  19. Final simplification0.4

    \[\leadsto \left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)}}\right) - 4.5\]

Reproduce

herbie shell --seed 2020066 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))